Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group.

used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II_25,1 in terms of the Leech lattice.

Description of the algorithm

Let $\Gamma < \mathrm(\mathbb^n)$ be a hyperbolic reflection group. Choose any point $v_0 \in \mathbb^n$; we shall call it the basic (or initial) point. The fundamental domain $P_0$ of its stabilizer $\Gamma_$ is a polyhedral cone in $\mathbb^n$.
Let $H_1,...,H_m$ be the faces of this cone, and let $a_1,...,a_m$ be outer normal vectors to it. Consider the half-spaces $H_k^- = \.$

There exists a unique fundamental polyhedron $P$ of $\Gamma$ contained in $P_0$ and containing the point $v_0$. Its faces containing $v_0$ are formed by faces $H_1,...,H_m$ of the cone $P_0$. The other faces $H_,...$ and the corresponding outward normals $a_, ...$ are constructed by induction. Namely, for $H_j$ we take a mirror such that the root $a_j$ orthogonal to it satisfies the conditions

(1) $(v_0,a_j) < 0$;

(2) $(a_i, a_j ) \le 0$ for all $i < j$;

(3) the distance $(v_0 , H_j)$ is minimum subject to constraints (1) and (2).

References

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*{{Citation , last1=Vinberg , first1=È. B. , editor1-last=Baily , editor1-first=Walter L. , title=Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) , url=https://books.google.com/books?id=7_g_AQAAIAAJ , publisher=Oxford University Press , isbn=978-0-19-560525-9 , mr=0422505 , year=1975 , chapter=Some arithmetical discrete groups in Lobačevskiĭ spaces , pages=323–348

Hyperbolic geometry
Reflection groups